When particle number is not fixed, relativity becomes essential or interactions create and destroy quanta, the wavefunction formalism must be replaced by quantum fields. Fields are operator-valued distributions acting on Fock space.

For a scalar field $\phi(x)$, the canonical momentum is $\pi(x)=\partial\mathcal{L}/\partial\dot\phi(x)$. Equal-time commutators such as $[\phi(x),\pi(y)]=i\hbar\,\delta(x-y)$ generalize particle quantization to infinitely many degrees of freedom.

The transition amplitude can be written as a sum over histories weighted by $e^{\frac{i}{\hbar}S}$. In Euclidean time, the path integral becomes a powerful tool for statistical mechanics, tunneling, and nonperturbative analysis.

Time-ordered correlation functions encode response and propagation. Perturbation theory organizes them through diagrams, while exact or numerical treatments often focus directly on correlators and spectral functions.

Quantum field theories often require renormalization: bare parameters are related to measurable quantities at a scale $\mu$. The renormalization group explains how couplings flow with scale and why effective theories have domain-dependent validity.